A well-known result of J. P. Serre states that foran arbitrary pair of points on a closed Riemannian manifoldthere exist infinitely many geodesics connecting these points.

A natural question is whether one can estimate the length of the"k-th" geodesic in terms of the diameter of a manifold.

We will demonstrate that given any pair of points p, q on a closedRiemannian manifold of dimension n and diameter d, there always exist atleast k geodesics of length at most 4nk^2d connecting them.

We will also demonstrate that for any two points of a manifold that isdiffeomorphic to the 2-sphere there always exist at least k geodesicsbetween them of length at most 24kd. (Joint with A. Nabutovsky)